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RD Chapter 3 Binary Operations Ex 3.3 Solutions

Question - 1 : -

Find the identity element in the set I+┬аof all positiveintegers defined by a * b = a + b for all a, b тИИ I+.

Answer - 1 : -

Let e be the identityelement in I+┬аwith respect to * such that

a * e = a = e * a, тИА a тИИ I+

a * e = a and e * a =a, тИА a тИИ I+

a + e = a and e + a =a, тИА a тИИ I+

e = 0, тИА a тИИ I+

Thus, 0 is theidentity element in I+┬аwith respect to *.

Question - 2 : -

Find the identity element in the set of all rational numbers except тАУ 1with respect to * defined by a * b = a + b + ab

Answer - 2 : -

Let e be the identityelement in I+┬аwith respect to * such that

a * e = a = e * a, тИА a тИИ Q тАУ {-1}

a * e = a and e * a =a, тИА a тИИ Q тАУ {-1}

a + e + ae = a and e +a + ea = a, тИА a тИИ Q тАУ {-1}

e + ae = 0 and e + ea= 0, тИА a тИИ Q тАУ {-1}

e (1 + a) = 0 and e (1+ a) = 0, тИА a тИИ Q тАУ {-1}

e = 0, тИА a тИИ Q тАУ {-1} [because a not equal to -1]

Thus, 0 is the identityelement in Q тАУ {-1} with respect to *.

Question - 3 : -

Answer - 3 : -


Question - 4 : -

Answer - 4 : -


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